Skip to main content
Article thumbnail
Location of Repository

Consistent order selection with strongly dependent data and its application to efficient estimation

By Javier Hidalgo


Order selection based on criteria by Akaike (1974), AIC, Schwarz (1978), BIC or Hannan and Quinn (1979) HIC is often applied in empirical examples. They have been used in the context of order selection of weakly dependent ARMA models, AR models with unit or explosive roots and in the context of regression or distributed lag regression models for weakly dependent data. On the other hand, it has been observed that data exhibits the so-called strong dependence in many areas. Because of the interest in this type of data, our main objective in this paper is to examine order selection for a distributed lag regression model that covers in a unified form weak and strong dependence. To that end, and because of the possible adverse properties of the aforementioned criteria, we propose a criterion function based on the decomposition of the variance of the innovations of the model in terms of their frequency components. Assuming that the order of the model is finite, say po , we show that the proposed criterion consistently estimates, po. In addition, we show that adaptive estimation for the parameters of the model is possible without knowledge of po . Finally, a small Monte-Carlo experiment is included to illustrate the finite sample performance of the proposed criterion

Topics: HB Economic Theory
Publisher: Suntory and Toyota International Centres for Economics and Related Disciplines, London School of Economics and Political Science
Year: 2002
OAI identifier:
Provided by: LSE Research Online

Suggested articles


  1. (5.6) We begin showing (5.5), whose left side is upper bounded by Pr
  2. (1956). A Combinatorial Lemma and its Application to Probability Theory,” doi
  3. (1979). a small Monte-Carlo study was carried out. The penalty functions g(T) were g(T)=l o gT and g(T)= 2.0log(logT), corresponding to those employed in the criteria BIC and HIC respectively.
  4. (2001). Adapting to Unknown Error Autocorrelation in Regression with Long Memory,” Forthcoming in Econometrica. doi
  5. (1993). An AIC Type Estimator for the Number of Cosinusoids,” doi
  6. (1980). An Introduction to Long Memory Time Series and Fractional Differencing,” doi
  7. (1981). An Optimal Selection of Regression Variables,” doi
  8. (1980). Asymptotic Efficiency Selection of the Order of the Model for Estimating Parameters of a Linear Process,” doi
  9. (1994). Determining the Number of Terms in a Trigonometric Regression,” doi
  10. (1979). Distributed Lag Approximation to Linear TimeInvariant Systems,” doi
  11. (1974). Distributed Lags,” doi
  12. (1981). Estimating Regression Models of Finite but doi
  13. (1978). Estimating the Dimension of a Model,” doi
  14. (1994). For a review of these models, see for instance Beran’s
  15. (1995). Gaussian Semiparametric Estimation of LongRange Dependence,” doi
  16. in (1.2) and (1.3), respectively, was used, and NP1 and NP2, the estimator of p0 when the criterion function used was S (p) or S∗ (p), given in (3.3) or (3.4),w i t hM =
  17. (2000). It is worth giving some intuition behind the criterion S (p). Since by Proposition 3 and Lemma 1 of Hidalgo
  18. (1989). Model Selection under Nonstationarity: Autoregressive Models and Stochastic Linear Regression Models,” doi
  19. (1997). Motivated by the last observation, Robinson and Hidalgo
  20. (1980). n dw h e r eΘ(λ) and Φ(λ) are the MA and AR polynomials respectively, having no zeroes in or on the unit circle. This is the familiar ARFIMA model, see for instance Granger and Joyeux
  21. (2000). Nonparametric Test for Causality with Long-Range Dependence,” doi
  22. (1998). On Unified Model Selection for Stationary and Nonstationary Short- and Long-Memory Autoregressive Processes,” doi
  23. (1974). Order selection based on criteria by Akaike
  24. (2001). Order Selection with Strongly Dependent doi
  25. (1963). Regression for Time Series,” doi
  26. (1976). Selection of the Order of an Autoregressive Model by Akaike’s Information Criterion,” doi
  27. (1994). Statistics for Long-Memory Processes. Chapman and Hall. doi
  28. (1974). t=1 wte itλ !Ã T X t=1 w 0 te −itλ ! (2.5) as the periodogram of wt. The estimator e θj in (2.3) was coined by Sims
  29. (1979). The Determination of the Order of an
  30. (1967). The Estimation of a Lagged Regression Relation,” doi
  31. (1991). The proof follows as a consequence of Theorem 3.3 and Lemma 1 of Pötscher
  32. (2001). The proof follows immediately from Hidalgo and Robinson’s
  33. (1974). The proof that Pr{b p∗ 0} → 0, follows by similar steps to those of Theorem 3.1, and it is omitted.
  34. the second term on the right of (5.8) is uniform in k.( R e c a l l t h a t t h e statistical properties of b θk are independent of k and/or the number of lags assumed in (1.1).) So, we obtain that (5.7) is upper bounded by Pr
  35. (2002). The Suntory Centre Suntory and Toyota International Centres for Economics and Related Disciplines London School of Economics and Political Science Discussion Paper Houghton Street No.EM/02/430 London WC2A 2AE
  36. (2000). The Variable Selection Problem,” doi
  37. (2000). Theorem 1 of Hidalgo
  38. (1997). Time Series Regression with Long Range Dependence,” doi
  39. (1994). Time Series with Strong Dependence,” doi
  40. (1981). Time Series, Data Analysis and Theory.S a n Francisco: Holden-Day. doi
  41. We begin showing that Pr{b p>p 0} → 0.D e n o t eb p = p0 + j for some j>0. Using the inequality Pr{b p>p 0} ≤
  42. (1981). We now elaborate on the results of Theorem 3.1. First, it is worth noting that unlike Geweke and Meese
  43. (1956). when p is allowed to increase with T,a si so u rc a s e ,(1.3) has some additional technical problems. In this situation, the proof of (1.5), and the route a d o p t e di nG e w e k ea n dM e e s e(1981), is based on Theorem 4.1 of Spitzer
  44. (1963). where fyx(λ) and fxx (λ) are the indicated elements of fww (λ) in (2.1).D u e t o this interpretation of θj,

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.